The Architecture of Music

Chords

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Rootless Chords, Inversions, and Symmetry

Although not included in the chord encyclopedia or incorporated with the scales portion of the book, all the rootless 3-tone chords and their inversions have been documented here for your exploration. When combined with the root position chords and their inversions on the previous page, the chords represent all 55 mathematically possible unique 3-tone chords. If you study the interval diagrams of the rootless chords you will find no perfect 5th interval exist in any of them. Since these chords have no perfect 5th interval they are considered rootless. In rootless chord interval diagrams, the circle simply indicates the 1st note or bass note, the X indicates the 2nd note, and triangle indicates the 3rd note as generated by the 3-tone chord matrix.

While a couple of these chords have names, most of them do not. The only record I could find of rootless chords were the augmented triad and diminished triad. The nameless rootless chords were simply named by their intervals. I’m sure some of these chords have names and are useful but more research into them needs to be done so they can be properly documented and organized. They have been included here at a minimum (formula wise) and in the book and can be explored using the generic piano and fretboard diagrams with the intervals for every bass note labeled provided in the book.

Rootless Chords, Inversions, and Negative Chords Diagram

Rootless Chords and Inversions